Constructive formalization of the Tennenbaum theorem and its applications

Plisko, V. E.

doi:10.1007/bf01157440

Cited by 7 publications

(14 citation statements)

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“…where T1 and U are the primitive recursive predicate and function, respectively, considered in the Kleene theorem on the normal form of partially recursive functions [11, w It is easy to show that in this case all the axioms of the system HAIS(A, B) are deducible in the system CHA described in [9]. The consistency of the latter with respect to HA can be proved by using realizability (see [10, 3.2…”

Section: Y(t(= Y)au(y) # 0) Y(t(z Y)au(y) = 0)mentioning

confidence: 98%

“…The system T = differs from the system CHA 2 constructed in [9] in that its postulates do not include the Markov principle M and the formal Church thesis CT. The latter is replaced by the axiom IS(A, B) included in the purely arithmetical part of the syste m T 2 .…”

Section: Tt) In F By F(tlt)= T and E(tlt2) By Tx = T2mentioning

confidence: 99%

“…For brevity, denote this theory by T. Let us put it through the procedure of eliminating function symbols and introducing predicate symbols in their place as described in [11, w that is, perform the same operations over T as were applied to the system HA in [9]. As a result, we obtain a system T' with the property that any arithmetic formula F satisfies the relation T~-F ,', '., T'}-F I, where F' is the predicate image of the formula F. Now, as in the construction of the system CHA 2 in [9], let us construct a formal system T 2 in the language which is the union of the languages of the systems T and T I .…”

Section: 22 (Ii)])mentioning

confidence: 99%

“…w Intuitionistic arithmetic in all finite types We shall consider a language or formal arithmetic containing symbols for all primitively recursive functions such that the symbol that designates a given primitively recursive function encodes its primitively recursive description. A version of such a language is presented in [9]. This paper also gives a description of the formal system of intuitioni, tic arithmetic HA whose postulates are the axiom schemes and inference rules of intuitionistic predicate calculus, the equality axioms, the Peano axioms, and the defining axioms for primitively recursive functions.…”

Section: W Introductionmentioning

confidence: 98%

“…The formalization of the Tennenbaum theorem is carried out in w Conceptually simple, this result is cumbersome technically. Its detailed exposition can be found in [8]. Here we only give a general scheme of the proof for certain statements.…”

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Modified realizability and predicate logic

Plisko

1997

Math Notes

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ABSTRACT. Semantics of predicate formulas based on the notion of modified realizability for arithmetic formulas and interpretations of the language of arithmetic in all finite types are considered. For a number of natural constructive interpretations, the corresponding predicate logic of modified realizability is proved to be nonarithmetical.KEY WORDS: modified realizability, arithmetic in all finite types, constructive logic.w IntroductionThe notion of recursive reaiizability for arithmetic formulas introduced by Kleene in [1] proved to be a useful tool in the study of intuitionistic theories. This notion can also be regarded as a special kind of semantics of arithmetic sentences. It is this semantics that underlies the constructive trend in mathematics developed by A. A. Markov and his school [2].In connection with the development of constructive mathematics, it is of interest to describe the logical laws admissible from the constructive point of view. The greatest progress in the description of the laws of constructive propositional logic was achieved by Varpakhovskii [3], who created a calculus in which all realizable propositional formulas known to date are deducible. Whether this calculus is complete remains an open question. The predicate logic of recursive realizability turned out to be rather complicated: it is not enumerable and even not arithmetical (see [4,5]).Recently, interest in constructive semantics has grown in connection with program synthesis. It is known that an intuitionistic proof of an arithmetic sentence of the form Vx 3yP(x, y) can be reworked into an algorithm that for any rn constructs an n such that P(m, n) holds. Constructive semantics of the arithmetic language are commonly used to derive algorithms from intuitionistic proofs. A number of constructive semantics can be obtained by means of a suitable generalization of recursive realizability. One of these generalizations is the modified realizability introduced by Kreisel [6]. It is a translation from the language of first order arithmetic into that of arithmetic in all finite types, which associates to each formula F a specially defined formula mr F in the language of arithmetic in all finite types. Having chosen some interpretation of the language of arithmetic in all finite types, we can define a semantics of arithmetic formulas by assuming a closed arithmetic formula F to be true (or M-realizable) if the formula mr F is true in the interpretation M.We can consider the predicate logic corresponding to the semantics thus obtained, that is, the collection of the predicate formulas all of whose arithmetic instances are true (M-realizable) for this semantics. These predicate formulas will also be called M-realizable. The aim of this paper is to show that for a number of natural interpretations of M the set of all M-realizable predicate formulas is not arithmetical. The proof of this fact is based on a formalization of the Tennenbaum theorem [7] according to which recursive nonstandard models of arithmetic are impossible.In w we des...

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Section: Y(t(= Y)au(y) # 0) Y(t(z Y)au(y) = 0)mentioning

confidence: 98%

Section: Tt) In F By F(tlt)= T and E(tlt2) By Tx = T2mentioning

confidence: 99%

Section: 22 (Ii)])mentioning

confidence: 99%

Section: W Introductionmentioning

confidence: 98%

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Modified realizability and predicate logic

Plisko

1997

Math Notes

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Arithmetic complexity of the predicate logics of certain complete arithmetic theories

Plisko

2001

Annals of Pure and Applied Logic

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Rules and Arithmetics

Visser¹

1999

Notre Dame J. Formal Logic

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This paper is concerned with the logical structure of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible propositional rules of Intuitionistic Propositional Logic. We provide some further insights concerning predicate logical admissible rules for arithmetical theories.

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Constructive formalization of the Tennenbaum theorem and its applications

Cited by 7 publications

References 2 publications

Modified realizability and predicate logic

Modified realizability and predicate logic

Arithmetic complexity of the predicate logics of certain complete arithmetic theories

Rules and Arithmetics

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